<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "https://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/xhtml;charset=UTF-8"/>
<meta http-equiv="X-UA-Compatible" content="IE=9"/>
<meta name="generator" content="Doxygen 1.9.1"/>
<meta name="viewport" content="width=device-width, initial-scale=1"/>
<title>Calico: calico/geometry.h Source File</title>
<link href="tabs.css" rel="stylesheet" type="text/css"/>
<script type="text/javascript" src="jquery.js"></script>
<script type="text/javascript" src="dynsections.js"></script>
<link href="search/search.css" rel="stylesheet" type="text/css"/>
<script type="text/javascript" src="search/searchdata.js"></script>
<script type="text/javascript" src="search/search.js"></script>
<script type="text/x-mathjax-config">
  MathJax.Hub.Config({
    extensions: ["tex2jax.js"],
    jax: ["input/TeX","output/HTML-CSS"],
});
</script>
<script type="text/javascript" async="async" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js"></script>
<link href="doxygen.css" rel="stylesheet" type="text/css" />
</head>
<body>
<div id="top"><!-- do not remove this div, it is closed by doxygen! -->
<div id="titlearea">
<table cellspacing="0" cellpadding="0">
 <tbody>
 <tr style="height: 56px;">
  <td id="projectalign" style="padding-left: 0.5em;">
   <div id="projectname">Calico
   </div>
   <div id="projectbrief">A visual-inertial calibration library designed for rapid problem construction and debugging.</div>
  </td>
 </tr>
 </tbody>
</table>
</div>
<!-- end header part -->
<!-- Generated by Doxygen 1.9.1 -->
<script type="text/javascript">
/* @license magnet:?xt=urn:btih:cf05388f2679ee054f2beb29a391d25f4e673ac3&amp;dn=gpl-2.0.txt GPL-v2 */
var searchBox = new SearchBox("searchBox", "search",false,'Search','.html');
/* @license-end */
</script>
<script type="text/javascript" src="menudata.js"></script>
<script type="text/javascript" src="menu.js"></script>
<script type="text/javascript">
/* @license magnet:?xt=urn:btih:cf05388f2679ee054f2beb29a391d25f4e673ac3&amp;dn=gpl-2.0.txt GPL-v2 */
$(function() {
  initMenu('',true,false,'search.php','Search');
  $(document).ready(function() { init_search(); });
});
/* @license-end */</script>
<div id="main-nav"></div>
<!-- window showing the filter options -->
<div id="MSearchSelectWindow"
     onmouseover="return searchBox.OnSearchSelectShow()"
     onmouseout="return searchBox.OnSearchSelectHide()"
     onkeydown="return searchBox.OnSearchSelectKey(event)">
</div>

<!-- iframe showing the search results (closed by default) -->
<div id="MSearchResultsWindow">
<iframe src="javascript:void(0)" frameborder="0" 
        name="MSearchResults" id="MSearchResults">
</iframe>
</div>

<div id="nav-path" class="navpath">
  <ul>
<li class="navelem"><a class="el" href="dir_e7ac338090e46d594d3ea7a0e9efc7ba.html">calico</a></li>  </ul>
</div>
</div><!-- top -->
<div class="header">
  <div class="headertitle">
<div class="title">geometry.h</div>  </div>
</div><!--header-->
<div class="contents">
<div class="fragment"><div class="line"><a name="l00001"></a><span class="lineno">    1</span>&#160;<span class="preprocessor">#ifndef CALICO_GEOMETRY_H_</span></div>
<div class="line"><a name="l00002"></a><span class="lineno">    2</span>&#160;<span class="preprocessor">#define CALICO_GEOMETRY_H_</span></div>
<div class="line"><a name="l00003"></a><span class="lineno">    3</span>&#160; </div>
<div class="line"><a name="l00004"></a><span class="lineno">    4</span>&#160;<span class="preprocessor">#include &quot;calico/typedefs.h&quot;</span></div>
<div class="line"><a name="l00005"></a><span class="lineno">    5</span>&#160;<span class="preprocessor">#include &quot;Eigen/Dense&quot;</span></div>
<div class="line"><a name="l00006"></a><span class="lineno">    6</span>&#160; </div>
<div class="line"><a name="l00007"></a><span class="lineno">    7</span>&#160; </div>
<div class="line"><a name="l00008"></a><span class="lineno">    8</span>&#160;<span class="keyword">namespace </span><a class="code" href="namespacecalico.html">calico</a> {</div>
<div class="line"><a name="l00009"></a><span class="lineno">    9</span>&#160; </div>
<div class="line"><a name="l00011"></a><span class="lineno">   11</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> T&gt;</div>
<div class="line"><a name="l00012"></a><span class="lineno"><a class="line" href="namespacecalico.html#a1bc1ca70bd6cfc6753dfef1ab61adff4">   12</a></span>&#160;<span class="keyword">inline</span> Eigen::Matrix3&lt;T&gt; <a class="code" href="namespacecalico.html#a1bc1ca70bd6cfc6753dfef1ab61adff4">Skew</a>(<span class="keyword">const</span> Eigen::Vector3&lt;T&gt;&amp; v) {</div>
<div class="line"><a name="l00013"></a><span class="lineno">   13</span>&#160;  Eigen::Matrix3&lt;T&gt; V;</div>
<div class="line"><a name="l00014"></a><span class="lineno">   14</span>&#160;  V.setZero();</div>
<div class="line"><a name="l00015"></a><span class="lineno">   15</span>&#160;  V(0, 1) = -v.z();</div>
<div class="line"><a name="l00016"></a><span class="lineno">   16</span>&#160;  V(1, 0) = v.z();</div>
<div class="line"><a name="l00017"></a><span class="lineno">   17</span>&#160;  V(0, 2) = v.y();</div>
<div class="line"><a name="l00018"></a><span class="lineno">   18</span>&#160;  V(2, 0) = -v.y();</div>
<div class="line"><a name="l00019"></a><span class="lineno">   19</span>&#160;  V(1, 2) = -v.x();</div>
<div class="line"><a name="l00020"></a><span class="lineno">   20</span>&#160;  V(2, 1) = v.x();</div>
<div class="line"><a name="l00021"></a><span class="lineno">   21</span>&#160;  <span class="keywordflow">return</span> V;</div>
<div class="line"><a name="l00022"></a><span class="lineno">   22</span>&#160;}</div>
<div class="line"><a name="l00023"></a><span class="lineno">   23</span>&#160; </div>
<div class="line"><a name="l00025"></a><span class="lineno">   25</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> T&gt;</div>
<div class="line"><a name="l00026"></a><span class="lineno"><a class="line" href="namespacecalico.html#a68a61002300d80b0d403719b982400eb">   26</a></span>&#160;<span class="keyword">inline</span> Eigen::Vector3&lt;T&gt; <a class="code" href="namespacecalico.html#a68a61002300d80b0d403719b982400eb">iSkew</a>(<span class="keyword">const</span> Eigen::Matrix3&lt;T&gt;&amp; V) {</div>
<div class="line"><a name="l00027"></a><span class="lineno">   27</span>&#160;  Eigen::Vector3&lt;T&gt; v;</div>
<div class="line"><a name="l00028"></a><span class="lineno">   28</span>&#160;  v.x() = V(2, 1) - V(1, 2);</div>
<div class="line"><a name="l00029"></a><span class="lineno">   29</span>&#160;  v.y() = V(0, 2) - V(2, 0);</div>
<div class="line"><a name="l00030"></a><span class="lineno">   30</span>&#160;  v.z() = V(1, 0) - V(0, 1);</div>
<div class="line"><a name="l00031"></a><span class="lineno">   31</span>&#160;  <span class="keywordflow">return</span> 0.5 * v;</div>
<div class="line"><a name="l00032"></a><span class="lineno">   32</span>&#160;}</div>
<div class="line"><a name="l00033"></a><span class="lineno">   33</span>&#160; </div>
<div class="line"><a name="l00035"></a><span class="lineno">   35</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> T&gt;</div>
<div class="line"><a name="l00036"></a><span class="lineno"><a class="line" href="namespacecalico.html#a98705a3b05fcd2103fa6823528c9cc7d">   36</a></span>&#160;<span class="keyword">inline</span> T <a class="code" href="namespacecalico.html#a98705a3b05fcd2103fa6823528c9cc7d">SmallAngleSin</a>(<span class="keyword">const</span> T theta) {</div>
<div class="line"><a name="l00037"></a><span class="lineno">   37</span>&#160;  <span class="keyword">const</span> T theta_sq = theta * theta;</div>
<div class="line"><a name="l00038"></a><span class="lineno">   38</span>&#160;  <span class="keywordflow">return</span> theta * (T(1.0) - theta_sq *</div>
<div class="line"><a name="l00039"></a><span class="lineno">   39</span>&#160;                  (T(1.0 / 6.0) + theta_sq *</div>
<div class="line"><a name="l00040"></a><span class="lineno">   40</span>&#160;                   (T(1.0 / 120.0) - theta_sq * T(1.0 / 5040.0))));</div>
<div class="line"><a name="l00041"></a><span class="lineno">   41</span>&#160;}</div>
<div class="line"><a name="l00042"></a><span class="lineno">   42</span>&#160; </div>
<div class="line"><a name="l00044"></a><span class="lineno">   44</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> T&gt;</div>
<div class="line"><a name="l00045"></a><span class="lineno"><a class="line" href="namespacecalico.html#a68a6ed102c3feb3b52f5b9950a6b2bc4">   45</a></span>&#160;<span class="keyword">inline</span> T <a class="code" href="namespacecalico.html#a68a6ed102c3feb3b52f5b9950a6b2bc4">SmallAngleCos</a>(<span class="keyword">const</span> T theta) {</div>
<div class="line"><a name="l00046"></a><span class="lineno">   46</span>&#160;  <span class="keyword">const</span> T theta_sq = theta * theta;</div>
<div class="line"><a name="l00047"></a><span class="lineno">   47</span>&#160;  <span class="keywordflow">return</span> T(1.0) - theta_sq * (T(0.5) - theta_sq *</div>
<div class="line"><a name="l00048"></a><span class="lineno">   48</span>&#160;                              (T(1.0 / 24.0) + theta_sq *</div>
<div class="line"><a name="l00049"></a><span class="lineno">   49</span>&#160;                               (T(1.0/720.0) - theta_sq * T(1.0 / 40320.0))));</div>
<div class="line"><a name="l00050"></a><span class="lineno">   50</span>&#160;}</div>
<div class="line"><a name="l00051"></a><span class="lineno">   51</span>&#160; </div>
<div class="line"><a name="l00053"></a><span class="lineno">   53</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> T&gt;</div>
<div class="line"><a name="l00054"></a><span class="lineno"><a class="line" href="namespacecalico.html#aa4a4bab45cfce99938f0b261e5464547">   54</a></span>&#160;<span class="keyword">inline</span> Eigen::Matrix3&lt;T&gt; <a class="code" href="namespacecalico.html#aa4a4bab45cfce99938f0b261e5464547">ExpSO3</a>(<span class="keyword">const</span> Eigen::Vector3&lt;T&gt;&amp; phi) {</div>
<div class="line"><a name="l00055"></a><span class="lineno">   55</span>&#160;  constexpr T kSmallAngle = 1e-7;</div>
<div class="line"><a name="l00056"></a><span class="lineno">   56</span>&#160;  <span class="keyword">const</span> T theta = phi.norm();</div>
<div class="line"><a name="l00057"></a><span class="lineno">   57</span>&#160;  <span class="keywordflow">if</span> (theta == T(0)) {</div>
<div class="line"><a name="l00058"></a><span class="lineno">   58</span>&#160;    <span class="keywordflow">return</span> Eigen::Matrix3&lt;T&gt;::Identity();</div>
<div class="line"><a name="l00059"></a><span class="lineno">   59</span>&#160;  }</div>
<div class="line"><a name="l00060"></a><span class="lineno">   60</span>&#160;  T sin_theta, one_m_cos_theta;</div>
<div class="line"><a name="l00061"></a><span class="lineno">   61</span>&#160;  <span class="keywordflow">if</span> (theta &lt; kSmallAngle) {</div>
<div class="line"><a name="l00062"></a><span class="lineno">   62</span>&#160;    sin_theta = <a class="code" href="namespacecalico.html#a98705a3b05fcd2103fa6823528c9cc7d">SmallAngleSin</a>(theta);</div>
<div class="line"><a name="l00063"></a><span class="lineno">   63</span>&#160;    one_m_cos_theta = T(1.0) - <a class="code" href="namespacecalico.html#a68a6ed102c3feb3b52f5b9950a6b2bc4">SmallAngleCos</a>(theta);</div>
<div class="line"><a name="l00064"></a><span class="lineno">   64</span>&#160;  } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00065"></a><span class="lineno">   65</span>&#160;    sin_theta = sin(theta);</div>
<div class="line"><a name="l00066"></a><span class="lineno">   66</span>&#160;    one_m_cos_theta = T(1.0) - cos(theta);</div>
<div class="line"><a name="l00067"></a><span class="lineno">   67</span>&#160;  }</div>
<div class="line"><a name="l00068"></a><span class="lineno">   68</span>&#160;  <span class="keyword">const</span> Eigen::Vector3&lt;T&gt; phi_hat = phi / theta;</div>
<div class="line"><a name="l00069"></a><span class="lineno">   69</span>&#160;  <span class="keyword">const</span> Eigen::Matrix3&lt;T&gt; Phi = <a class="code" href="namespacecalico.html#a1bc1ca70bd6cfc6753dfef1ab61adff4">Skew</a>(phi_hat);</div>
<div class="line"><a name="l00070"></a><span class="lineno">   70</span>&#160;  Eigen::Matrix3&lt;T&gt; R =</div>
<div class="line"><a name="l00071"></a><span class="lineno">   71</span>&#160;      Eigen::Matrix3&lt;T&gt;::Identity() + sin_theta * Phi +</div>
<div class="line"><a name="l00072"></a><span class="lineno">   72</span>&#160;      one_m_cos_theta * Phi * Phi;</div>
<div class="line"><a name="l00073"></a><span class="lineno">   73</span>&#160;  <span class="keywordflow">return</span> R;</div>
<div class="line"><a name="l00074"></a><span class="lineno">   74</span>&#160;}</div>
<div class="line"><a name="l00075"></a><span class="lineno">   75</span>&#160; </div>
<div class="line"><a name="l00078"></a><span class="lineno">   78</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> T&gt;</div>
<div class="line"><a name="l00079"></a><span class="lineno"><a class="line" href="namespacecalico.html#a4b5e83bfad1cc6b767ae5354d5d483c8">   79</a></span>&#160;<span class="keyword">inline</span> Eigen::Vector3&lt;T&gt; <a class="code" href="namespacecalico.html#a4b5e83bfad1cc6b767ae5354d5d483c8">LnSO3</a>(<span class="keyword">const</span> Eigen::Matrix3&lt;T&gt;&amp; R) {</div>
<div class="line"><a name="l00080"></a><span class="lineno">   80</span>&#160;  constexpr T kInvSqrt2 = T(1.0 / sqrt(2.0));</div>
<div class="line"><a name="l00081"></a><span class="lineno">   81</span>&#160;  constexpr T kSmallAngle = 1e-7;</div>
<div class="line"><a name="l00082"></a><span class="lineno">   82</span>&#160;  <span class="keyword">const</span> T tr = R.trace();</div>
<div class="line"><a name="l00083"></a><span class="lineno">   83</span>&#160;  <span class="keywordflow">if</span> (tr == T(3.0)) {</div>
<div class="line"><a name="l00084"></a><span class="lineno">   84</span>&#160;    <span class="keywordflow">return</span> Eigen::Vector3&lt;T&gt;::Zero();</div>
<div class="line"><a name="l00085"></a><span class="lineno">   85</span>&#160;  }</div>
<div class="line"><a name="l00086"></a><span class="lineno">   86</span>&#160;  Eigen::Vector3&lt;T&gt; phi = <a class="code" href="namespacecalico.html#a68a61002300d80b0d403719b982400eb">iSkew</a>(R);</div>
<div class="line"><a name="l00087"></a><span class="lineno">   87</span>&#160;  <span class="keyword">const</span> T cos_theta = T(0.5) * (tr - T(1.0));</div>
<div class="line"><a name="l00088"></a><span class="lineno">   88</span>&#160;  <span class="keyword">const</span> T sin_theta = phi.norm();</div>
<div class="line"><a name="l00089"></a><span class="lineno">   89</span>&#160; </div>
<div class="line"><a name="l00090"></a><span class="lineno">   90</span>&#160;  <span class="keywordflow">if</span> (cos_theta &gt;= kInvSqrt2) {</div>
<div class="line"><a name="l00091"></a><span class="lineno">   91</span>&#160;    <span class="keyword">const</span> T theta = asin(sin_theta);</div>
<div class="line"><a name="l00092"></a><span class="lineno">   92</span>&#160;    <span class="keyword">const</span> T sin_theta = theta &lt; kSmallAngle ? <a class="code" href="namespacecalico.html#a98705a3b05fcd2103fa6823528c9cc7d">SmallAngleSin</a>(theta) : sin(theta);</div>
<div class="line"><a name="l00093"></a><span class="lineno">   93</span>&#160;    phi *= (theta / sin_theta);</div>
<div class="line"><a name="l00094"></a><span class="lineno">   94</span>&#160;  } <span class="keywordflow">else</span> <span class="keywordflow">if</span> (cos_theta &gt; -kInvSqrt2) {</div>
<div class="line"><a name="l00095"></a><span class="lineno">   95</span>&#160;    <span class="keyword">const</span> T theta = acos(cos_theta);</div>
<div class="line"><a name="l00096"></a><span class="lineno">   96</span>&#160;    <span class="keyword">const</span> T sin_theta = theta &lt; kSmallAngle ? <a class="code" href="namespacecalico.html#a98705a3b05fcd2103fa6823528c9cc7d">SmallAngleSin</a>(theta) : sin(theta);</div>
<div class="line"><a name="l00097"></a><span class="lineno">   97</span>&#160;    phi *= (theta / sin_theta);</div>
<div class="line"><a name="l00098"></a><span class="lineno">   98</span>&#160;  } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00099"></a><span class="lineno">   99</span>&#160;    <span class="keyword">const</span> Eigen::Vector3&lt;T&gt; diag = R.diagonal().array() - cos_theta;</div>
<div class="line"><a name="l00100"></a><span class="lineno">  100</span>&#160;    <span class="keyword">const</span> T dx2 = diag.x() * diag.x();</div>
<div class="line"><a name="l00101"></a><span class="lineno">  101</span>&#160;    <span class="keyword">const</span> T dy2 = diag.y() * diag.y();</div>
<div class="line"><a name="l00102"></a><span class="lineno">  102</span>&#160;    <span class="keyword">const</span> T dz2 = diag.z() * diag.z();</div>
<div class="line"><a name="l00103"></a><span class="lineno">  103</span>&#160;    Eigen::Vector3&lt;T&gt; axis;</div>
<div class="line"><a name="l00104"></a><span class="lineno">  104</span>&#160;    <span class="keywordflow">if</span> ((dx2 &gt; dz2) &amp;&amp; (dx2 &gt; dy2)){</div>
<div class="line"><a name="l00105"></a><span class="lineno">  105</span>&#160;      axis.x() = diag.x();</div>
<div class="line"><a name="l00106"></a><span class="lineno">  106</span>&#160;      axis.y() = T(0.5) * (R(0, 1) + R(1, 0));</div>
<div class="line"><a name="l00107"></a><span class="lineno">  107</span>&#160;      axis.z() = T(0.5) * (R(0, 2) + R(2, 0));</div>
<div class="line"><a name="l00108"></a><span class="lineno">  108</span>&#160;    } <span class="keywordflow">else</span> <span class="keywordflow">if</span> (dy2 &gt; dz2) {</div>
<div class="line"><a name="l00109"></a><span class="lineno">  109</span>&#160;      axis.x() = T(0.5) * (R(1, 0) + R(0, 1));</div>
<div class="line"><a name="l00110"></a><span class="lineno">  110</span>&#160;      axis.y() = diag.y();</div>
<div class="line"><a name="l00111"></a><span class="lineno">  111</span>&#160;      axis.z() = T(0.5) * (R(1, 2) + R(2, 1));</div>
<div class="line"><a name="l00112"></a><span class="lineno">  112</span>&#160;    } <span class="keywordflow">else</span>{</div>
<div class="line"><a name="l00113"></a><span class="lineno">  113</span>&#160;      axis.x() = T(0.5) * (R(2, 0) + R(0, 2));</div>
<div class="line"><a name="l00114"></a><span class="lineno">  114</span>&#160;      axis.y() = T(0.5) * (R(2, 1) + R(1, 2));</div>
<div class="line"><a name="l00115"></a><span class="lineno">  115</span>&#160;      axis.z() = diag.z();</div>
<div class="line"><a name="l00116"></a><span class="lineno">  116</span>&#160;    }</div>
<div class="line"><a name="l00117"></a><span class="lineno">  117</span>&#160;    <span class="keywordflow">if</span> ((phi.dot(axis)) &lt; T(0)) {</div>
<div class="line"><a name="l00118"></a><span class="lineno">  118</span>&#160;      axis *= T(-1.0);</div>
<div class="line"><a name="l00119"></a><span class="lineno">  119</span>&#160;    }</div>
<div class="line"><a name="l00120"></a><span class="lineno">  120</span>&#160;    <span class="keyword">const</span> T theta = M_PI - asin(sin_theta);</div>
<div class="line"><a name="l00121"></a><span class="lineno">  121</span>&#160;    phi = theta * axis.normalized();</div>
<div class="line"><a name="l00122"></a><span class="lineno">  122</span>&#160;  }</div>
<div class="line"><a name="l00123"></a><span class="lineno">  123</span>&#160;  <span class="keywordflow">return</span> phi;</div>
<div class="line"><a name="l00124"></a><span class="lineno">  124</span>&#160;}</div>
<div class="line"><a name="l00125"></a><span class="lineno">  125</span>&#160; </div>
<div class="line"><a name="l00137"></a><span class="lineno">  137</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> T&gt;</div>
<div class="line"><a name="l00138"></a><span class="lineno"><a class="line" href="namespacecalico.html#a405ed3df9c4ed8eebcabe5ec92d6717e">  138</a></span>&#160;<span class="keyword">inline</span> Eigen::Matrix3&lt;T&gt; <a class="code" href="namespacecalico.html#a405ed3df9c4ed8eebcabe5ec92d6717e">ExpSO3Jacobian</a>(</div>
<div class="line"><a name="l00139"></a><span class="lineno">  139</span>&#160;    <span class="keyword">const</span> Eigen::Vector3&lt;T&gt;&amp; phi) {</div>
<div class="line"><a name="l00140"></a><span class="lineno">  140</span>&#160;  <span class="keyword">const</span> T theta_sq = phi.squaredNorm();</div>
<div class="line"><a name="l00141"></a><span class="lineno">  141</span>&#160;  Eigen::Matrix3&lt;T&gt; J;</div>
<div class="line"><a name="l00142"></a><span class="lineno">  142</span>&#160;  J.setIdentity();</div>
<div class="line"><a name="l00143"></a><span class="lineno">  143</span>&#160;  <span class="keywordflow">if</span> (theta_sq == <span class="keyword">static_cast&lt;</span>T<span class="keyword">&gt;</span>(0.0)) {</div>
<div class="line"><a name="l00144"></a><span class="lineno">  144</span>&#160;    <span class="keywordflow">return</span> J;</div>
<div class="line"><a name="l00145"></a><span class="lineno">  145</span>&#160;  }</div>
<div class="line"><a name="l00146"></a><span class="lineno">  146</span>&#160;  <span class="keyword">const</span> T theta = sqrt(theta_sq);</div>
<div class="line"><a name="l00147"></a><span class="lineno">  147</span>&#160;  T one_m_cos_theta, sin_theta;</div>
<div class="line"><a name="l00148"></a><span class="lineno">  148</span>&#160;  <span class="keywordflow">if</span> (theta &lt; T(1e-7)) {</div>
<div class="line"><a name="l00149"></a><span class="lineno">  149</span>&#160;    sin_theta = <a class="code" href="namespacecalico.html#a98705a3b05fcd2103fa6823528c9cc7d">SmallAngleSin</a>(theta);</div>
<div class="line"><a name="l00150"></a><span class="lineno">  150</span>&#160;    one_m_cos_theta = T(1.0) - <a class="code" href="namespacecalico.html#a68a6ed102c3feb3b52f5b9950a6b2bc4">SmallAngleCos</a>(theta);</div>
<div class="line"><a name="l00151"></a><span class="lineno">  151</span>&#160;  } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00152"></a><span class="lineno">  152</span>&#160;    sin_theta = sin(theta);</div>
<div class="line"><a name="l00153"></a><span class="lineno">  153</span>&#160;    one_m_cos_theta = T(1.0) - cos(theta);</div>
<div class="line"><a name="l00154"></a><span class="lineno">  154</span>&#160;  }</div>
<div class="line"><a name="l00155"></a><span class="lineno">  155</span>&#160;  <span class="keyword">const</span> T inv_theta = T(1.0) / theta;</div>
<div class="line"><a name="l00156"></a><span class="lineno">  156</span>&#160;  <span class="keyword">const</span> Eigen::Vector3&lt;T&gt; phi_hat = inv_theta * phi;</div>
<div class="line"><a name="l00157"></a><span class="lineno">  157</span>&#160;  <span class="keyword">const</span> Eigen::Matrix3&lt;T&gt; phi_hat_x = <a class="code" href="namespacecalico.html#a1bc1ca70bd6cfc6753dfef1ab61adff4">Skew</a>(phi_hat);</div>
<div class="line"><a name="l00158"></a><span class="lineno">  158</span>&#160;  J += inv_theta * (one_m_cos_theta * phi_hat_x +</div>
<div class="line"><a name="l00159"></a><span class="lineno">  159</span>&#160;                    (theta - sin_theta) * phi_hat_x * phi_hat_x);</div>
<div class="line"><a name="l00160"></a><span class="lineno">  160</span>&#160;  <span class="keywordflow">return</span> J;</div>
<div class="line"><a name="l00161"></a><span class="lineno">  161</span>&#160;}</div>
<div class="line"><a name="l00162"></a><span class="lineno">  162</span>&#160; </div>
<div class="line"><a name="l00172"></a><span class="lineno">  172</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> T&gt;</div>
<div class="line"><a name="l00173"></a><span class="lineno"><a class="line" href="namespacecalico.html#a3e0aabf168db953b2b5903af43105582">  173</a></span>&#160;<span class="keyword">inline</span> std::vector&lt;Eigen::Matrix3&lt;T&gt;&gt; <a class="code" href="namespacecalico.html#a3e0aabf168db953b2b5903af43105582">ExpSO3Hessian</a>(</div>
<div class="line"><a name="l00174"></a><span class="lineno">  174</span>&#160;    <span class="keyword">const</span> Eigen::Vector3&lt;T&gt;&amp; phi) {</div>
<div class="line"><a name="l00175"></a><span class="lineno">  175</span>&#160;  <span class="keyword">const</span> std::vector&lt;Eigen::Matrix3&lt;T&gt;&gt; G = {</div>
<div class="line"><a name="l00176"></a><span class="lineno">  176</span>&#160;    <a class="code" href="namespacecalico.html#a1bc1ca70bd6cfc6753dfef1ab61adff4">Skew</a>(Eigen::Vector3&lt;T&gt;(T(1), T(0), T(0))),</div>
<div class="line"><a name="l00177"></a><span class="lineno">  177</span>&#160;    <a class="code" href="namespacecalico.html#a1bc1ca70bd6cfc6753dfef1ab61adff4">Skew</a>(Eigen::Vector3&lt;T&gt;(T(0), T(1), T(0))),</div>
<div class="line"><a name="l00178"></a><span class="lineno">  178</span>&#160;    <a class="code" href="namespacecalico.html#a1bc1ca70bd6cfc6753dfef1ab61adff4">Skew</a>(Eigen::Vector3&lt;T&gt;(T(0), T(0), T(1)))</div>
<div class="line"><a name="l00179"></a><span class="lineno">  179</span>&#160;  };</div>
<div class="line"><a name="l00180"></a><span class="lineno">  180</span>&#160; </div>
<div class="line"><a name="l00181"></a><span class="lineno">  181</span>&#160;  std::vector&lt;Eigen::Matrix3&lt;T&gt;&gt; H(3);</div>
<div class="line"><a name="l00182"></a><span class="lineno">  182</span>&#160;  std::fill(H.begin(), H.end(), Eigen::Matrix3&lt;T&gt;::Zero());</div>
<div class="line"><a name="l00183"></a><span class="lineno">  183</span>&#160;  <span class="keyword">const</span> T theta_sq = phi.squaredNorm();</div>
<div class="line"><a name="l00184"></a><span class="lineno">  184</span>&#160;  <span class="keywordflow">if</span> (theta_sq == T(0.0)) {</div>
<div class="line"><a name="l00185"></a><span class="lineno">  185</span>&#160;    <span class="keywordflow">return</span> H;</div>
<div class="line"><a name="l00186"></a><span class="lineno">  186</span>&#160;  }</div>
<div class="line"><a name="l00187"></a><span class="lineno">  187</span>&#160;  <span class="keyword">const</span> T theta = sqrt(theta_sq);</div>
<div class="line"><a name="l00188"></a><span class="lineno">  188</span>&#160;  T ct, st;</div>
<div class="line"><a name="l00189"></a><span class="lineno">  189</span>&#160;  <span class="keywordflow">if</span> (theta &lt; T(1e-7)) {</div>
<div class="line"><a name="l00190"></a><span class="lineno">  190</span>&#160;    ct = <a class="code" href="namespacecalico.html#a68a6ed102c3feb3b52f5b9950a6b2bc4">SmallAngleCos</a>(theta);</div>
<div class="line"><a name="l00191"></a><span class="lineno">  191</span>&#160;    st = <a class="code" href="namespacecalico.html#a98705a3b05fcd2103fa6823528c9cc7d">SmallAngleSin</a>(theta);</div>
<div class="line"><a name="l00192"></a><span class="lineno">  192</span>&#160;  } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00193"></a><span class="lineno">  193</span>&#160;    ct = cos(theta);</div>
<div class="line"><a name="l00194"></a><span class="lineno">  194</span>&#160;    st = sin(theta);</div>
<div class="line"><a name="l00195"></a><span class="lineno">  195</span>&#160;  }</div>
<div class="line"><a name="l00196"></a><span class="lineno">  196</span>&#160;  <span class="keyword">const</span> T inv_theta = T(1.0) / theta;</div>
<div class="line"><a name="l00197"></a><span class="lineno">  197</span>&#160;  <span class="keyword">const</span> T inv_theta_sq = inv_theta * inv_theta;</div>
<div class="line"><a name="l00198"></a><span class="lineno">  198</span>&#160;  <span class="keyword">const</span> Eigen::Vector3&lt;T&gt; phi_hat = inv_theta * phi;</div>
<div class="line"><a name="l00199"></a><span class="lineno">  199</span>&#160;  <span class="keyword">const</span> Eigen::Matrix3&lt;T&gt; phi_hat_x = <a class="code" href="namespacecalico.html#a1bc1ca70bd6cfc6753dfef1ab61adff4">Skew</a>(phi_hat);</div>
<div class="line"><a name="l00200"></a><span class="lineno">  200</span>&#160;  <span class="keyword">const</span> T c0 = ct - st * inv_theta;</div>
<div class="line"><a name="l00201"></a><span class="lineno">  201</span>&#160;  <span class="keyword">const</span> T c1 = (T(1.0) - ct) * inv_theta_sq;</div>
<div class="line"><a name="l00202"></a><span class="lineno">  202</span>&#160;  <span class="keyword">const</span> T c2 = T(3.0) * inv_theta_sq * st - inv_theta * (ct - T(2.0));</div>
<div class="line"><a name="l00203"></a><span class="lineno">  203</span>&#160;  <span class="keyword">const</span> T c3 = inv_theta_sq * (theta - st);</div>
<div class="line"><a name="l00204"></a><span class="lineno">  204</span>&#160;  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; 3; ++i) {</div>
<div class="line"><a name="l00205"></a><span class="lineno">  205</span>&#160;    H[i] = (c0 * phi_hat(i)) * phi_hat_x + c1 * G[i] +</div>
<div class="line"><a name="l00206"></a><span class="lineno">  206</span>&#160;        (c2 * phi_hat(i)) * (phi_hat_x * phi_hat_x) +</div>
<div class="line"><a name="l00207"></a><span class="lineno">  207</span>&#160;        c3 * (G[i] * phi_hat_x + phi_hat_x * G[i]);</div>
<div class="line"><a name="l00208"></a><span class="lineno">  208</span>&#160;  }</div>
<div class="line"><a name="l00209"></a><span class="lineno">  209</span>&#160;  <span class="keywordflow">return</span> H;</div>
<div class="line"><a name="l00210"></a><span class="lineno">  210</span>&#160;}</div>
<div class="line"><a name="l00211"></a><span class="lineno">  211</span>&#160; </div>
<div class="line"><a name="l00213"></a><span class="lineno">  213</span>&#160;<span class="keyword">template</span> &lt;<span class="keyword">typename</span> T&gt;</div>
<div class="line"><a name="l00214"></a><span class="lineno"><a class="line" href="namespacecalico.html#a7d85e76032e1f2ab3267b64b3992b0d3">  214</a></span>&#160;<span class="keyword">inline</span> Eigen::Matrix3&lt;T&gt; <a class="code" href="namespacecalico.html#a7d85e76032e1f2ab3267b64b3992b0d3">ExpSO3JacobianDot</a>(</div>
<div class="line"><a name="l00215"></a><span class="lineno">  215</span>&#160;    <span class="keyword">const</span> Eigen::Vector3&lt;T&gt;&amp; phi, <span class="keyword">const</span> Eigen::Vector3&lt;T&gt;&amp; phi_dot) {</div>
<div class="line"><a name="l00216"></a><span class="lineno">  216</span>&#160;  <span class="keyword">const</span> std::vector&lt;Eigen::Matrix3&lt;T&gt;&gt; H = <a class="code" href="namespacecalico.html#a3e0aabf168db953b2b5903af43105582">ExpSO3Hessian</a>(phi);</div>
<div class="line"><a name="l00217"></a><span class="lineno">  217</span>&#160;  Eigen::Matrix3&lt;T&gt; Jdot;</div>
<div class="line"><a name="l00218"></a><span class="lineno">  218</span>&#160;  <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 0; i &lt; H.size(); ++i) {</div>
<div class="line"><a name="l00219"></a><span class="lineno">  219</span>&#160;    Jdot.col(i) = H.at(i) * phi_dot;</div>
<div class="line"><a name="l00220"></a><span class="lineno">  220</span>&#160;  }</div>
<div class="line"><a name="l00221"></a><span class="lineno">  221</span>&#160;  <span class="keywordflow">return</span> Jdot;</div>
<div class="line"><a name="l00222"></a><span class="lineno">  222</span>&#160;}</div>
<div class="line"><a name="l00223"></a><span class="lineno">  223</span>&#160; </div>
<div class="line"><a name="l00224"></a><span class="lineno">  224</span>&#160;} <span class="comment">// namespace calico</span></div>
<div class="line"><a name="l00225"></a><span class="lineno">  225</span>&#160; </div>
<div class="line"><a name="l00226"></a><span class="lineno">  226</span>&#160;<span class="preprocessor">#endif </span><span class="comment">// CALICO_GEOMETRY_H_</span></div>
<div class="ttc" id="anamespacecalico_html"><div class="ttname"><a href="namespacecalico.html">calico</a></div><div class="ttdoc">Primary calico namespace.</div><div class="ttdef"><b>Definition:</b> __init__.py:1</div></div>
<div class="ttc" id="anamespacecalico_html_a1bc1ca70bd6cfc6753dfef1ab61adff4"><div class="ttname"><a href="namespacecalico.html#a1bc1ca70bd6cfc6753dfef1ab61adff4">calico::Skew</a></div><div class="ttdeci">Eigen::Matrix3&lt; T &gt; Skew(const Eigen::Vector3&lt; T &gt; &amp;v)</div><div class="ttdoc">Convert a 3-vector to a skew-symmetric cross-product matrix.</div><div class="ttdef"><b>Definition:</b> geometry.h:12</div></div>
<div class="ttc" id="anamespacecalico_html_a3e0aabf168db953b2b5903af43105582"><div class="ttname"><a href="namespacecalico.html#a3e0aabf168db953b2b5903af43105582">calico::ExpSO3Hessian</a></div><div class="ttdeci">std::vector&lt; Eigen::Matrix3&lt; T &gt; &gt; ExpSO3Hessian(const Eigen::Vector3&lt; T &gt; &amp;phi)</div><div class="ttdef"><b>Definition:</b> geometry.h:173</div></div>
<div class="ttc" id="anamespacecalico_html_a405ed3df9c4ed8eebcabe5ec92d6717e"><div class="ttname"><a href="namespacecalico.html#a405ed3df9c4ed8eebcabe5ec92d6717e">calico::ExpSO3Jacobian</a></div><div class="ttdeci">Eigen::Matrix3&lt; T &gt; ExpSO3Jacobian(const Eigen::Vector3&lt; T &gt; &amp;phi)</div><div class="ttdef"><b>Definition:</b> geometry.h:138</div></div>
<div class="ttc" id="anamespacecalico_html_a4b5e83bfad1cc6b767ae5354d5d483c8"><div class="ttname"><a href="namespacecalico.html#a4b5e83bfad1cc6b767ae5354d5d483c8">calico::LnSO3</a></div><div class="ttdeci">Eigen::Vector3&lt; T &gt; LnSO3(const Eigen::Matrix3&lt; T &gt; &amp;R)</div><div class="ttdef"><b>Definition:</b> geometry.h:79</div></div>
<div class="ttc" id="anamespacecalico_html_a68a61002300d80b0d403719b982400eb"><div class="ttname"><a href="namespacecalico.html#a68a61002300d80b0d403719b982400eb">calico::iSkew</a></div><div class="ttdeci">Eigen::Vector3&lt; T &gt; iSkew(const Eigen::Matrix3&lt; T &gt; &amp;V)</div><div class="ttdoc">Converts a skew symmetric matrix back into a vector.</div><div class="ttdef"><b>Definition:</b> geometry.h:26</div></div>
<div class="ttc" id="anamespacecalico_html_a68a6ed102c3feb3b52f5b9950a6b2bc4"><div class="ttname"><a href="namespacecalico.html#a68a6ed102c3feb3b52f5b9950a6b2bc4">calico::SmallAngleCos</a></div><div class="ttdeci">T SmallAngleCos(const T theta)</div><div class="ttdoc">Fourth order Taylor expansion of cos(theta).</div><div class="ttdef"><b>Definition:</b> geometry.h:45</div></div>
<div class="ttc" id="anamespacecalico_html_a7d85e76032e1f2ab3267b64b3992b0d3"><div class="ttname"><a href="namespacecalico.html#a7d85e76032e1f2ab3267b64b3992b0d3">calico::ExpSO3JacobianDot</a></div><div class="ttdeci">Eigen::Matrix3&lt; T &gt; ExpSO3JacobianDot(const Eigen::Vector3&lt; T &gt; &amp;phi, const Eigen::Vector3&lt; T &gt; &amp;phi_dot)</div><div class="ttdoc">Compute the time derivative of the Rodrigues formula manifold Jacobian.</div><div class="ttdef"><b>Definition:</b> geometry.h:214</div></div>
<div class="ttc" id="anamespacecalico_html_a98705a3b05fcd2103fa6823528c9cc7d"><div class="ttname"><a href="namespacecalico.html#a98705a3b05fcd2103fa6823528c9cc7d">calico::SmallAngleSin</a></div><div class="ttdeci">T SmallAngleSin(const T theta)</div><div class="ttdoc">Fourth order Taylor expansion of sin(theta).</div><div class="ttdef"><b>Definition:</b> geometry.h:36</div></div>
<div class="ttc" id="anamespacecalico_html_aa4a4bab45cfce99938f0b261e5464547"><div class="ttname"><a href="namespacecalico.html#aa4a4bab45cfce99938f0b261e5464547">calico::ExpSO3</a></div><div class="ttdeci">Eigen::Matrix3&lt; T &gt; ExpSO3(const Eigen::Vector3&lt; T &gt; &amp;phi)</div><div class="ttdoc">Convert axis-angle to rotation matrix.</div><div class="ttdef"><b>Definition:</b> geometry.h:54</div></div>
</div><!-- fragment --></div><!-- contents -->
<!-- start footer part -->
<hr class="footer"/><address class="footer"><small>
Generated by&#160;<a href="https://www.doxygen.org/index.html"><img class="footer" src="doxygen.svg" width="104" height="31" alt="doxygen"/></a> 1.9.1
</small></address>
</body>
</html>
